# Examples why university education is important for future high school teachers

At my university, the students in math are mixed up (1/3-1/2 are bachelor/master students, the rest are future high school teachers). A problem arising very often is the discussion dramatically summarized by "I don't need this in high school/I already know enough, don't bother me with this abstract stuff".

Are there good examples/showcases/questions/arguments/etc. illustrating how important math education at university is for their life as high school teachers?

(Expect from general discussion like "You need to know more than the school kids/What if they change the content of high school education/What will you tell the school kids when the say that they wanted to study something completely unrelated to math?" - If you want to, you can also add general issues, but I am more interetesed in concrete examples)

• If I recall correctly, it's not. For example, teachers with Masters degrees don't perform better than those with just Bachelors degrees. Knowing the abstract stuff doesn't really help them teach the high school stuff, and in any case you can easily get a degree in math education without taking an upper level proof-based math class. – Potato Mar 14 '14 at 17:57
• @Thomas: Sorry, no. I am looking for issues like "Without learning about <university subject>, it is difficult to understand the background of <school subject>". Or some arguments like "Only if you saw (at some point) the edge of reaearch based questions you really know that mathematics is and can teach" – Markus Klein Mar 14 '14 at 18:44
• I'm confused by the first sentence. The high school teachers don't need a bachelor/masters degree? – JDH Mar 27 '14 at 22:31
• @JDH No, they don't need it, but (at least) in German universities, most of the courses in mathematics are for both, bachelor candidates and high school teacher candidates. – Markus Klein Mar 27 '14 at 22:40
• I know this question is old, but wanted to note (for future readers, even if the OP knows this well) that this can be country-dependent. The life of a "high-school teacher" can be rather different in Western Europe (non-UK), Eastern Europe, the UK, Canada, the USA... – Yemon Choi May 26 '16 at 2:25

I'm not completely sure how well this addresses the question, but here is my best response. A few years ago I was teaching a Methods course for preservice secondary math teachers. Over the course of the semester there was a fair amount of griping, not about my course but about the other courses they were required to take, and in particular about their Abstract Algebra course, which they felt was completely useless for their future careers.

At a certain point in the semester we were doing some collaborative lesson planning for a hypothetical set of classes about exponential functions. Since our Methods class was structured in part around the "backwards design" approach of Wiggins & McTighe, we spent some time at the beginning discussing the "big ideas" of exponential functions. One of my students proposed as a "big idea" that there was an analogy between exponential growth and linear growth: in the functions $$f(x)=mx+b$$ and $$g(x)=AB^x$$, the coefficient $$A$$ is analogous to the constant term $$b$$ (both give the y-intercept or "starting value" for the function) and the exponential base $$B$$ is analogous to the slope $$m$$ (both capture the idea of a "constant growth rate", one additively and one multiplicatively).

After talking about this for a while I said something like: "You know, there is a precise name for 'analogies' like this. A mathematician would call it an isomorphism. In fact what you seem to be saying is that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. In fact the properties of logarithms and exponents — like $$e^{a+b} = e^a e^b$$ and $$\log(ab)=\log(a) + \log(b)$$ — really don't say anything more than that the log and exp functions are homomorphisms."

One of my students got one of those "aha" looks and commented that I had just taught her more about abstract algebra in 2 minutes than she learned in an entire semester of the other course.

(An even more basic version of this: Students already know that the rules for multiplying positive and negative numbers are, in a sense, "just like" the rules for adding even and odd numbers. Abstract algebra provides a language for describing exactly what this "just like" sense means.)

My point is that there are ways in which higher-level undergraduate math courses (like abstract algebra) can be directly relevant for teaching secondary mathematics... But also that those applications are usually not made explicit. So students are probably not wrong in feeling that those courses are useless for their future careers. It is up to us to connect the dots and show them how and why these things matter.

Edited, November 2020:. I hope it is not considered shilling to add this here, but in the six years since I originally wrote this post I have written and published a mathematics textbook that is explicitly organized around the goal I articulated in the last paragraph: "to connect the dots [between advanced level mathematics and the secondary curriculum] and show them [i.e. preservice and/or inservice secondary teachers] how and why these things matter". The book is called Secondary Mathematics for Mathematicians and Educators, and is published by Routledge. (More information available at the link in the previous sentence.)

• I think that isomorphism example is exactly on target: You can tell people about it's definition all day long, they won't understand a word (because they don't see the point); show an example, all becomes clear. – vonbrand Mar 16 '14 at 16:02
• I used to use the same analogy in comparing arithmetic series and geometric series. Also, suppose you want to find values of $A$ and $B$ such that the graph of $y = AB^x$ passes through two given points, a common problem in precalculus and one that students have difficulty with. One way of solving it is to write the two equations, then divide the equations ($A$ drops out), and I would motivate this approach by showing that it's the exact analogy of fitting a line $y = mx + b$ to two given points by writing the appropriate equations and then subtracting the equations $(b$ drops out). – Dave L Renfro Apr 11 '14 at 17:36
• It's bizarre that the students were complaining about abstract algebra, which is directly related to teaching math if anything is. Things like why the product of two negative numbers is positive, associative commutative and distributive properties, and of course isomorphisms. Sad that so many people fail to know or learn how to apply it correctly in teaching. – 6005 Apr 12 '14 at 16:39
• Popping back in here after six years to respond to @6005's comment: I really don't think the problem is the students. It's a real mistake to think that these connections are obvious, and that anybody who doesn't see them is at fault. On the contrary the whole point of my example is that these connections are only obvious to us because we have a perspective that lets us look back and see them. That's why it is up to us to make those connections visible for students (as I wrote originally, to "connect the dots and show them how and why these things matter"). – mweiss Nov 19 '20 at 0:25

Here is some text from the paper "Teacher Characteristics and Student Achievement Gains: A Review" (Wayne and Youngs, 2003):

The question is this: "why do you want to teach math ?"

I'll try to stay positive here, I think anyone who has had experience with potential teachers is well aware that bad teachers are probably first bad students. Moreover, as one of my colleagues said, difference between doctors and us: they bury theirs we just graduate them.

That aside, I would like for them to be motivated by genuine curiosity for math. If they don't have it already then comments ought to be made to shame them into caring. What are numbers? Why does algebra work like it does? Why do we need calculus? What if you did trigonometry with some shape other than a circle? I read in this book that you can cut a volume into two pieces with the same volume, what's the deal with that? What does it mean to be in 4-dimensions? Why are functions important, who thought of them first?

These are just some examples, but, I think we can agree that a student will be much more likely to respect you as a teacher if you can at least start down a path which tells them how they might want to think to answer such questions. In other words, they ought to study abstraction and origins with some fervor. If your best answer is for them to google it, why bother teaching? As teachers of math we need to know math. It's just that simple.

• This is a reason why society should require high school math teachers to have a university math education. However, the OP described the context, which is that the future teachers are resistant to learning math, and the OP wants a good argument to be able to give them. You're essentially saying that these people should be prevented from becoming teachers because they're bad students. That's probably true, but it's not an argument that can be effective when presented to them. – Ben Crowell Jul 15 at 12:53

I have faced exactly the same issue. I often teach masters level math courses to students getting a masters in education with a focus in mathematics; so these are future high-school math teachers.

The way I look at it, my real task is to provide them with mathematical enrichment and sophistication, to show them a larger context and provide a deeper perspective from which they might look upon the more elementary material with a greater degree of mathematical insight.

But indeed, they will not directly be teaching this more sophisticated material in their high-school classes. Nevertheless, I do believe that the enrichment I offer is indeed enriching, and develops important parts of their mathematical thinking, such as a greater appreciation for mathematical proof and a greater facility with abstract thinking. And this I believe is in turn beneficial for their own students.

Faced with the occasional objection that they don't need that stuff (and it is usually only the weaker students who make such objections), I have found that I can almost always convince them simply by asking:

Everyone agrees, of course, that a 2nd grade (age 7) reading teacher should read at far better than 3rd grade (age 8) level. And similarly, for the same obvious reasons, a math teacher at any level should know far more than just the material they are teaching their students.

• I like the wording enriching and add mathematical sophistication to future math teachers, not just rushing them for the knowledge sake. – Ming Lei Jan 2 '18 at 4:39

Of course, as already mentioned, it is important for a teacher to know more than her or his students. But one thing where this principle is especially important is exercise (and test) designing. Let me give a few simple examples.

1. How do teachers manage to chose the second degree polynoms that have not-too-complicated roots? They start from the factorization, which needs a slightly deeper understanding of polynoms than root finding methods.
2. How can a teacher improvise a $2\times 2$ affine system with a unique solution (or with no or multiple solutions) at the board, whenever needed (e.g. some students finished all exercises already)? They use determinant. Variant: finding $3$ affine equations on $2$ variables that have solutions by using row manipulations, which needs some understanding of independence in linear algebra.
3. We know how to compute sequences defined by rank-two induction by reducing the corresponding matrix. This can be used to design a problem (e.g. on Fibonacci sequence) where the relevant change of variable is given to the students.

There are obviously a whole lot of such examples, where deeper understanding of somehow more abstract mathematics than taught is needed for teachers simply to design good exercises.

How can you hope to clearly explain something you yourself do not understand well?

I have found the following line of reasoning is quite convincing for many skeptics:

How hard did precalculus seem when you took it? After taking a semester of calculus, how did your perception change? How about after taking 3 semesters of calculus?

Mathematics is cumulative. The more you learn, the more you reinforce the basics.

By the time you finish the standard sequence of several semesters of calculus, differential equations and linear algebra, you start to find that algebra and the basic precalculus material is equally difficult with basic arithmetic. When first confronted with derivatives, maybe they seemed weird/difficult/tricky, but after a couple of semesters of material building on the basics, taking a derivative is like adding 2 and 2 or breathing.

The point is, the more you know, the more you will consider basic/easy/trivial. When teaching a subject, you want to be able to focus on how you are presenting the material - how to best communicate this to you students. You don't want to be fighting with understanding it yourself.

Along the same lines, the more you know, the better perspective you have as to why different topics are important. The more math you learn, the better you will be armed when confronted with a skeptical student of your own who asks, "Why do I need to learn this?"

Admittedly modern algebra as taught at most universities is a hard sell for you typical unmotivated future highschool teacher (why you would enter such a profession half-hearted, I do not know, but it seems common). If you focus on ring theory, you have a chance at selling the connections to factorization and the like. Groups are trickier because, honestly, that specific material may very well be completely unapplicable. However, students who spend a semester really engaging their modern algebra (or any other proof oriented) class should leave with at the very least bullet proof logic skills. This of course is of immeasurable value when teaching mathematics.

Anyway, that's how I've approached this topic with many students (and been successful as far as I can tell).

To teach mathematics, you need to know, first, what mathematics is, no exceptions.

The problem of school-math being a whole different thing having little in common with mathematics, is caused, among others, by people who teach school-math, but have a bad idea about what mathematics is. However, it's a hard chance that someone will have contact with mathematics in high school, in fact, if some student does, with high probability he/she will go to college (perhaps with major other than math, but still).

This might change in the future, but right know, the best option to learn mathematics is to go to the academia. Of course, just attending university courses is not enough, but itself. However, there are opportunities, many times more than somewhere else.

To this end, I would gladly require any math teacher to do some research (I'm not suggesting that teachers should teach research in high school, only that knowing what research looks like in math gives you an appropriate perspective). It might be very small and insignificant, but still. On the other hand, I understand this is highly unfeasible, there aren't enough people that would be able to do it, after all, nowadays you can do a PhD without having an independent thought $\ddot\frown$

So, what's left is hope that going to some university will be enough. It's not encouraging and I wouldn't ever use such an argument to convince some future teacher (the effect might be just the opposite). Nevertheless, it is a real concern in present times.

To conclude: going to universities is the future teachers' best chance to get acquainted with mathematics.

I hope this helps $\ddot\smile$

• As a Math Ed student and general math nerd, the general distaste towards deep math by math teachers at the high school level is extremely upsetting. I am hoping to eventually get my PhD in Math, but not because I want to teach at the college level... I just think it would be neat to learn all the maths that's known currently. – David G Mar 15 '14 at 7:08
• @Skytso I need to warn you that it's practically impossible to know all the maths that's known currently. You might want to learn the most important things from each domain, but to know it all - I think it would be hard even to read all the theses and papers that are published each year. – dtldarek Mar 15 '14 at 8:53
• @dtldarek: That comment is a masterpiece of understatement. – Mark Meckes Mar 15 '14 at 12:43
• To be not that understating, I would say that it is even impossible to understand all the major problems – let alone all the answers. – Wrzlprmft Mar 16 '14 at 19:07
• A kid can dream, right? Sorry for taking this discussion off course. =P – David G Mar 28 '14 at 0:48

While teaching polynomials to future high-school teachers, I realized one example of pretty advanced mathematics that is needed to understand why a certain important (but often kept implicit) result in high school mathematics is a result, and not something so obvious that it need not be even stated:

Two real polynomials taking the same value at each point are equal coefficient-wise.

It has proved hugely difficult to make my students realize there where something to state here, because they did not question the notion of equality, and did not realize that the definition of equality for functions and for polynomials might conflict each other.

Now, once one has seen finite fields, one can be given an example of a non-zero polynomial whose polynomial function vanishes identically. I don't see any better way to show that there is something to prove in the real case. There are some tricks, such as using trigonometric polynomials for comparison ($\cos^2+\sin^2$ is coefficient-wise different from $1$).

But the finite-field example made it also possible for me to convince my student that they would never, ever be able to prove that coefficient-wise equality and functional equality agree just by using algebraic manipulations, because they would be able to apply their proof to the finite field case. This helped them accepting that it is worth giving a proof, but also that this proof has to use something more than the field operations (I gave two flavor of proofs, one using polynomial algebra and the fact that there are infinitely many real numbers, and one using topological arguments).

Of course, as high-school teachers they will not teach finite fields, and will probably not insist on the distinction between equality as polynomial and functions. But they have learned that there is something they will be pushing under the rug, making them certainly better at presenting the material, understanding some possible misconceptions by students, and answering some questions if they turn out. These look like invaluable teaching skills.

• Can you spell out your topological argument? I think I know what "polynomial algebra" proof you have in mind. A third proof follows the development of Taylor's formula: evaluate derivatives of all orders at $x=0$. – Steven Gubkin Nov 19 '20 at 2:58
• @StevenGubkin The argument was simply to compute the limits in $\pm\infty$ of the polynomial function, if I recall correctly. This might be considered analytic more than topological, I guess. – Benoît Kloeckner Nov 19 '20 at 8:08
• Taking limits seems unsatisfying. If you're willing to use calculus in the proof, then there are simpler things to do, like differentiating until you get a constant polynomial whose value is zero but whose constant is nonzero. The elementary argument would I think be that if a polynomial $P$ is identically zero, then its constant coefficient must be zero, so you can factor out an x, giving $P=xQ$. Then you have to use the fact that nonzero reals have nonzero products. – Ben Crowell Jul 15 at 13:21

When my high school offered the AP course in Calculus, the teacher chosen to teach the class was the one who had studied it most recently. She was rusty and it showed. She was continually making mistakes in her demonstrations and the students wound up frequently correcting her.

When I tutored college algebra, I frequently encountered prospective elementary teachers who did not like or understand mathematics, and suspected that their attitudes and lack of understanding would be reflected in their teaching. The same would be of course be true in high-school teachers.

• It is my firm belief that everyone who dislikes a subject has a teacher to blame at some point. If you dislike math, you should not teach it or your students will learn your dislike as well as the content. – David G Mar 15 '14 at 7:04
• @Skytso Sometimes it's enough to have a parent who dislikes that subject. – dtldarek Mar 16 '14 at 21:40

Examples from Abstract Algebra

• The reason why the product of two negative numbers is positive boils down to a proof from abstract algebra. $(-3)(4)$ should be the opposite of $(3)(4)$ because you add the two to get zero. Then $(-3)(-4)$ should in turn be the opposite of $(-3)(4)$.

• Why negative numbers exist, and why subtraction is just adding a negative.

• Why fractions exist.

• Why zero behaves oddly with multiplication and division.

• Why $-1$ is not a prime number. Why $1$ is not a prime number.

• Why $\infty$ is often not considered a number.

• What we mean in general when we talk about a "number".

• $\cdots$

With all of the above questions, I think only a trained mathematician would know how to answer the question honestly if they needed to explain it to a child.

I'm sure there are many examples from analysis too, but abstract algebra in particular is practically the direct study of everything taught in basic elementary school math: integers, addition and multiplication, divisibility, fractions, and so on.

This is but a part of the bigger picture and certainly not the most important one: Speeding up the legwork.

Obviously you have to be better than your students in what you teach to, e.g., design and correct tests in a reasonable time. There are two ways of getting better at such things: Getting more routine with the techniques you are teaching or having a deeper understanding and knowing more advanced techniques. The former has its limits (and so does the latter) and thus you need the latter, at least to some extent.

One illustrative example (that is unfortunately far from the high-school level and also not fully realistic):

Is the following calculation correct? $$152345+10340+532045+139525+10530+123525=968313$$

You can decide about this having learnt only how to add by simply calculating the correct result and checking it. But most likely you will quickly notice that the sum of some numbers ending on 5 and 0 can never end on 3. But that already requires more knowledge than just being able to add numbers mechanically.

• I agree with most of your argumentation (Speeding up is good - But this is maybe an argument against higher education since the students can then argue by "See: The best way to become a good teacher is to to do a lot of exercises on the level on which I will teach in the future and the learn some tricks"). I see the point of your example, but that is maybe something a student in higher class in high school can also see. I was more looking for examples where higher math is useful to teach high school students. Something in the spirit like in @mweiss answer. – Markus Klein Mar 16 '14 at 21:39

I don’t think that you can explain this to the students. Perhaps instead, you should better familiarize yourself with the high school standards and draw the connections.

A high school mathematics teacher will teach material that heavily overlaps with all undergraduate coursework, and is only a slight extension of most ideas.

While a high school mathematics teacher may spend their time teaching a bunch of 9th grade algebra, there is much to connect in their linear algebra and/or modern algebra courses.

They should also be prepared to teach through BC calculus or HL IB mathematics, which are not superficial. Further, AP stats requires a solid understanding of introductory statistics.

I would recommend these students work related to making some of these curricular connections and how they could be creative about teaching them. Incorporate the wealth of research on teaching and learning advanced mathematics, discuss math teams, research opportunities for high schoolers, etc.

High school teaching is a very demanding profession that requires deep understanding of formal mathematics at a high level of you are going to be a true professional and find continuing opportunities for yourself. There should be no argument about this, just look at what high school mathematics students are expected to do in terms of material content through precalc in the CCSS and the AP or IB exams.

I think the HS teachers have a point. It seems like you love the topic and want to find a justification for it (examples why the teachers are wrong) rather than really considering the question openly if maybe they are right. Maybe the reason you strain so hard for a rationale is the the math teachers have a point!

Consider: I like gymnastics as a sport, but it is a stretch to say that working on it will help a high school football player get better at football or be a sound use of his time. (Wrestling on the other hand, especially for defensive players has some noticeable benefit. But is still not a requirement.)

• The answer would be improved by arguing why advanced math is not useful for math teachers. Do you have, for example, studies to back it up? – Tommi Dec 28 '17 at 14:30
• I would put the burden of proof on the opposite. – guest Dec 28 '17 at 22:10
• This is not an argument; you are attempting to answer a question, but right now your answer is a simple statement with nothing but a weak analogy backing it up. This makes your answer a poor one, in my opinion. (There are also many other poor answers here, for the same reason, and some good ones.) – Tommi Dec 29 '17 at 8:43